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Stats › Backtesting and evidence › Drawdown maths

Recovery takes three times as long as the fall, whatever your Sharpe ratio

Sharpe 1.0Sharpe 1.5
Expected annual profit$10m$15m
Annual standard deviation$10m$10m
Maximum quantile loss at 95%$6,763,859$4,509,239
Time to reach it0.676 years0.300 years
Total time under water2.706 years1.202 years
Ratio of the two3 to 13 to 1

Two worked examples at 95% confidence with monthly observations, assuming independent normally distributed profit and loss. Raising the Sharpe ratio from 1.0 to 1.5 cuts the worst expected loss by a third and cuts the time under water by more than half, but it does not change the three to one ratio between recovery and decline.

For a strategy with a Sharpe ratio of 1, the worst expected loss at 95% confidence arrives after 0.68 years and the strategy stays under water for 2.7 years. The ratio is three to one, and under the model's assumptions it does not depend on the Sharpe ratio at all.

3 to 1Time under water against time to the worst loss, at any Sharpe ratio
$6.76mWorst expected loss at 95%, Sharpe 1.0, on $10m annual volatility
2.71 yearsTime under water in that case
1.20 yearsThe same for a Sharpe of 1.5

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